To analyze the solution(s) to the equation \( x^2 + 5x + 6 = -3x^2 - 2x \), we first rearrange it to one side of the equation:
\[ x^2 + 5x + 6 + 3x^2 + 2x = 0 \]
This simplifies to:
\[ 4x^2 + 7x + 6 = 0 \]
Next, we can factor or use the quadratic formula to find the values of \( x \) that would yield 0. First, let's factor the quadratic:
The equation \( 4x^2 + 7x + 6 \) factors as:
\[ (4x + 3)(x + 2) = 0 \]
Setting each factor to zero gives the solutions:
- \( 4x + 3 = 0 \) leads to \( x = -\frac{3}{4} \)
- \( x + 2 = 0 \) leads to \( x = -2 \)
So the solutions to the equation are \( x = -\frac{3}{4} \) and \( x = -2 \).
Now, let's carefully evaluate the statements:
A. An input of -3 will yield an output of 0 on both sides of the equation.
We check \( x = -3 \):
Left side: \[ (-3)^2 + 5(-3) + 6 = 9 - 15 + 6 = 0 \] Right side: \[ -3(-3)^2 - 2(-3) = -3(9) + 6 = -27 + 6 = -21 \quad (\text{not 0}) \]
This statement is false.
B. There are no input values that would yield the same output value on both sides of the equation.
Since we found solutions \( x = -\frac{3}{4} \) and \( x = -2 \), there are indeed input values that yield the same output (0). This statement is false.
C. An input of 0 will yield an output of 0 on both sides of the equation.
We check with \( x = 0 \):
Left side: \[ 0^2 + 5(0) + 6 = 6 \] Right side: \[ -3(0^2) - 2(0) = 0 \quad (\text{not equal}) \]
This statement is false.
D. An input of -2 will yield an output of 0 on both sides of the equation.
We check with \( x = -2 \):
Left side: \[ (-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0 \] Right side: \[ -3(-2)^2 - 2(-2) = -3(4) + 4 = -12 + 4 = -8 \quad (\text{not equal}) \]
This statement is false.
Given the evaluations, the only correct interpretation among the options provided, considering we found solutions, is that input values such as \( x = -2 \) yield an output of 0 for the left side, but not for the right side.
None of the statements provided were entirely accurate given the analysis. However, the correct values to satisfy the equation are indeed \( x = -\frac{3}{4} \) and \( x = -2 \). Thus, the option being most relevant is inconsistent given the calculations. The most reflective conclusion based on the statement focus should highlight that solutions do exist.
However, in a strict interpretation of what has been given, please go with statement identification as they logically appear, noting both aspects of output findings.